Properties

Label 12960.bq.1620.c1.a1
Order $ 2^{3} $
Index $ 2^{2} \cdot 3^{4} \cdot 5 $
Normal No

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Subgroup ($H$) information

Description:$C_2\times C_4$
Order: \(8\)\(\medspace = 2^{3} \)
Index: \(1620\)\(\medspace = 2^{2} \cdot 3^{4} \cdot 5 \)
Exponent: \(4\)\(\medspace = 2^{2} \)
Generators: $\langle(10,12)(11,14), (4,9), (4,9)(5,8)(10,14,12,11)\rangle$ Copy content Toggle raw display
Nilpotency class: $1$
Derived length: $1$

The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.

Ambient group ($G$) information

Description: $F_5\times S_3\wr C_3$
Order: \(12960\)\(\medspace = 2^{5} \cdot 3^{4} \cdot 5 \)
Exponent: \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \)
Derived length:$3$

The ambient group is nonabelian and monomial (hence solvable).

Automorphism information

While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.

$\operatorname{Aut}(G)$$F_5\times S_3\wr S_3$, of order \(25920\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5 \)
$\operatorname{Aut}(H)$ $D_4$, of order \(8\)\(\medspace = 2^{3} \)
$W$$C_1$, of order $1$

Related subgroups

Centralizer:$C_{12}:C_2^3$
Normalizer:$C_{12}:C_2^3$
Normal closure:$F_5\times S_3^3$
Core:$C_1$
Minimal over-subgroups:$C_2\times F_5$$C_2\times C_{12}$$C_6:C_4$$C_4\times S_3$$C_2^2\times C_4$$C_2^2\times C_4$$C_2^2\times C_4$
Maximal under-subgroups:$C_2^2$$C_4$$C_4$
Autjugate subgroups:12960.bq.1620.c1.a2

Other information

Number of subgroups in this conjugacy class$135$
Möbius function$0$
Projective image$F_5\times S_3\wr C_3$